I (and many contributors to this site) prefer to rent a car through a wholesale consolidator.
The company most often cited here is:
www.autoeurope.com and their sister company www.kemwel.com
They are part of the same group, but I check prices with both since the sometimes prices differ a bit and therefore I pick the better deal in either site.
They work with the largest rental companies in Europe, such as Avis, Budget, Europcar, Hertz, etc.
I also select the no deductible insurance option (a.k.a. ‘no insurance excess’) since my US car insurance does not cover me abroad. It’s a bit more expensive, but the deductible can easily amount to €1,000 or more. Also coverage provided by many credit cards excludes Italy and a handful of other countries.
I’ve rented through both consolidator websites above (generally selecting Europcar or Hertz as the rental provider, as they tend to have cheaper rates). When you return the car make sure you obtain receipt that you filled your tank (I also take a picture of the dashboard showing the full gasoline gauge). I often get charged for fuel charges after the fact when I’m back home and having those records allow me to dispute the extra charge with my credit card.
Don’t forget that to drive in Europe you need an International Driver’s Permit (obtainable in 10 minutes at any AAA office with $15 and 2 passport photos). The IDP is a translation of your home driver’s license in multiple languages.
I also recommend to take a GPS unit from home (Garmin or TomTom are both good albeit not infallible) with European maps loaded on it (Maps available on Amazon) and an old fashioned paper map for planning your itinerary and also as a back up since GPS navigators are sometimes inaccurate. You can buy a paper map locally at any shop that sells newspapers.
The beach is a good option for a 11 y.o. In particular there is a very famous beach near Porto Empedocle that is called "La Scala dei Turchi" or the "Staircase of the Turks". It's actually a cliff formation, not a beach. You should visit it, since it's very close to San Leone. Here is a video.
But of course you can't go to Sicily and not visit the many historical sights of a region that was settled by numerous civilizations present a different times in its extraordinarily rich history. Sicily has been under the Phoenicians, the Carthaginians, the Greeks, the Romans, the Arabs, the Normans, the Spaniards, and lastly the Italians. All of them have left their mark.
It takes about 2 hours from Noto to Mt Etna by car. Leave early because the Etna National Park is a whole day affair.